69 Search Results for "Saraf, Shubhangi"

Volume

LIPIcs, Volume 169

35th Computational Complexity Conference (CCC 2020)

CCC 2020, July 28-31, 2020, Saarbrücken, Germany (Virtual Conference)

Editors: Shubhangi Saraf

Document
DOI: 10.4230/LIPIcs.ESA.2024.89
Locally Computing Edge Orientations

Authors: Slobodan Mitrović, Ronitt Rubinfeld, and Mihir Singhal

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
We consider the question of orienting the edges in a graph G such that every vertex has bounded out-degree. For graphs of arboricity α, there is an orientation in which every vertex has out-degree at most α and, moreover, the best possible maximum out-degree of an orientation is at least α - 1. We are thus interested in algorithms that can achieve a maximum out-degree of close to α. A widely studied approach for this problem in the distributed algorithms setting is a "peeling algorithm" that provides an orientation with maximum out-degree α(2+ε) in a logarithmic number of iterations. We consider this problem in the local computation algorithm (LCA) model, which quickly answers queries of the form "What is the orientation of edge (u,v)?" by probing the input graph. When the peeling algorithm is executed in the LCA setting by applying standard techniques, e.g., the Parnas-Ron paradigm, it requires Ω(n) probes per query on an n-vertex graph. In the case where G has unbounded degree, we show that any LCA that orients its edges to yield maximum out-degree r must use Ω(√ n/r) probes to G per query in the worst case, even if G is known to be a forest (that is, α = 1). We also show several algorithms with sublinear probe complexity when G has unbounded degree. When G is a tree such that the maximum degree Δ of G is bounded, we demonstrate an algorithm that uses Δ n^{1-log_Δ r + o(1)} probes to G per query. To obtain this result, we develop an edge-coloring approach that ultimately yields a graph-shattering-like result. We also use this shattering-like approach to demonstrate an LCA which 4-colors any tree using sublinear probes per query.
Cite as

Slobodan Mitrović, Ronitt Rubinfeld, and Mihir Singhal. Locally Computing Edge Orientations. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 89:1-89:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mitrovic_et_al:LIPIcs.ESA.2024.89,
  author =	{Mitrovi\'{c}, Slobodan and Rubinfeld, Ronitt and Singhal, Mihir},
  title =	{{Locally Computing Edge Orientations}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{89:1--89:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.89},
  URN =		{urn:nbn:de:0030-drops-211603},
  doi =		{10.4230/LIPIcs.ESA.2024.89},
  annote =	{Keywords: local computation algorithms, edge orientation, tree coloring}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.51
Consequences of Randomized Reductions from SAT to Time-Bounded Kolmogorov Complexity

Authors: Halley Goldberg and Valentine Kabanets

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
A central open question within meta-complexity is that of NP-hardness of problems such as MCSP and MK^{t}P. Despite a large body of work giving consequences of and barriers for NP-hardness of these problems under (restricted) deterministic reductions, very little is known in the setting of randomized reductions. In this work, we give consequences of randomized NP-hardness reductions for both approximating and exactly computing time-bounded and time-unbounded Kolmogorov complexity. In the setting of approximate K^{poly} complexity, our results are as follows. 1) Under a derandomization assumption, for any constant δ > 0, if approximating K^t complexity within n^{δ} additive error is hard for SAT under an honest randomized non-adaptive Turing reduction running in time polynomially less than t, then NP = coNP. 2) Under the same assumptions, the worst-case hardness of NP is equivalent to the existence of one-way functions. Item 1 above may be compared with a recent work of Saks and Santhanam [Michael E. Saks and Rahul Santhanam, 2022], which makes the same assumptions except with ω(log n) additive error, obtaining the conclusion NE = coNE. In the setting of exact K^{poly} complexity, where the barriers of Item 1 and [Michael E. Saks and Rahul Santhanam, 2022] do not apply, we show: 3) If computing K^t complexity is hard for SAT under reductions as in Item 1, then the average-case hardness of NP is equivalent to the existence of one-way functions. That is, "Pessiland" is excluded. Finally, we give consequences of NP-hardness of exact time-unbounded Kolmogorov complexity under randomized reductions. 4) If computing Kolmogorov complexity is hard for SAT under a randomized many-one reduction running in time t_R and with failure probability at most 1/(t_R)^16, then coNP is contained in non-interactive statistical zero-knowledge; thus NP ⊆ coAM. Also, the worst-case hardness of NP is equivalent to the existence of one-way functions. We further exploit the connection to NISZK along with a previous work of Allender et al. [Eric Allender et al., 2023] to show that hardness of K complexity under randomized many-one reductions is highly robust with respect to failure probability, approximation error, output length, and threshold parameter.
Cite as

Halley Goldberg and Valentine Kabanets. Consequences of Randomized Reductions from SAT to Time-Bounded Kolmogorov Complexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 51:1-51:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{goldberg_et_al:LIPIcs.APPROX/RANDOM.2024.51,
  author =	{Goldberg, Halley and Kabanets, Valentine},
  title =	{{Consequences of Randomized Reductions from SAT to Time-Bounded Kolmogorov Complexity}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{51:1--51:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.51},
  URN =		{urn:nbn:de:0030-drops-210444},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.51},
  annote =	{Keywords: Meta-complexity, Randomized reductions, NP-hardness, Worst-case complexity, Time-bounded Kolmogorov complexity}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.53
When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?

Authors: Dean Doron, Jonathan Mosheiff, and Mary Wootters

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
The Gilbert-Varshamov (GV) bound is a classical existential result in coding theory. It implies that a random linear binary code of rate ε² has relative distance at least 1/2 - O(ε) with high probability. However, it is a major challenge to construct explicit codes with similar parameters. One hope to derandomize the Gilbert-Varshamov construction is with code concatenation: We begin with a (hopefully explicit) outer code 𝒞_out over a large alphabet, and concatenate that with a small binary random linear code 𝒞_in. It is known that when we use independent small codes for each coordinate, then the result lies on the GV bound with high probability, but this still uses a lot of randomness. In this paper, we consider the question of whether code concatenation with a single random linear inner code 𝒞_in can lie on the GV bound; and if so what conditions on 𝒞_out are sufficient for this. We show that first, there do exist linear outer codes 𝒞_out that are "good" for concatenation in this sense (in fact, most linear codes codes are good). We also provide two sufficient conditions for 𝒞_out, so that if 𝒞_out satisfies these, 𝒞_out∘𝒞_in will likely lie on the GV bound. We hope that these conditions may inspire future work towards constructing explicit codes 𝒞_out.
Cite as

Dean Doron, Jonathan Mosheiff, and Mary Wootters. When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 53:1-53:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{doron_et_al:LIPIcs.APPROX/RANDOM.2024.53,
  author =	{Doron, Dean and Mosheiff, Jonathan and Wootters, Mary},
  title =	{{When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{53:1--53:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.53},
  URN =		{urn:nbn:de:0030-drops-210467},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.53},
  annote =	{Keywords: Error-correcting codes, Concatenated codes, Derandomization, Gilbert-Varshamov bound}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.61
When Can an Expander Code Correct Ω(n) Errors in O(n) Time?

Authors: Kuan Cheng, Minghui Ouyang, Chong Shangguan, and Yuanting Shen

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
Tanner codes are graph-based linear codes whose parity-check matrices can be characterized by a bipartite graph G together with a linear inner code C₀. Expander codes are Tanner codes whose defining bipartite graph G has good expansion property. This paper is motivated by the following natural and fundamental problem in decoding expander codes: What are the sufficient and necessary conditions that δ and d₀ must satisfy, so that every bipartite expander G with vertex expansion ratio δ and every linear inner code C₀ with minimum distance d₀ together define an expander code that corrects Ω(n) errors in O(n) time? For C₀ being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT'96) showed that δ > 3/4 is sufficient; later Viderman (ACM-TOCT'13) improved this to δ > 2/3-Ω(1) and he also showed that δ > 1/2 is necessary. For general linear code C₀, the previously best-known result of Dowling and Gao (IEEE-TIT'18) showed that d₀ = Ω(cδ^{-2}) is sufficient, where c is the left-degree of G. In this paper, we give a near-optimal solution to the above question for general C₀ by showing that δ d₀ > 3 is sufficient and δ d₀ > 1 is necessary, thereby also significantly improving Dowling-Gao’s result. We present two novel algorithms for decoding expander codes, where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius.
Cite as

Kuan Cheng, Minghui Ouyang, Chong Shangguan, and Yuanting Shen. When Can an Expander Code Correct Ω(n) Errors in O(n) Time?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 61:1-61:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cheng_et_al:LIPIcs.APPROX/RANDOM.2024.61,
  author =	{Cheng, Kuan and Ouyang, Minghui and Shangguan, Chong and Shen, Yuanting},
  title =	{{When Can an Expander Code Correct \Omega(n) Errors in O(n) Time?}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{61:1--61:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.61},
  URN =		{urn:nbn:de:0030-drops-210543},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.61},
  annote =	{Keywords: expander codes, expander graphs, linear-time decoding}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.69
Randomness Extractors in AC⁰ and NC¹: Optimal up to Constant Factors

Authors: Kuan Cheng and Ruiyang Wu

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
We study randomness extractors in AC⁰ and NC¹. For the AC⁰ setting, we give a logspace-uniform construction such that for every k ≥ n/poly log n, ε ≥ 2^{-poly log n}, it can extract from an arbitrary (n, k) source, with a small constant fraction entropy loss, and the seed length is O(log n/(ε)). The seed length and output length are optimal up to constant factors matching the parameters of the best polynomial time construction such as [Guruswami et al., 2009]. The range of k and ε almost meets the lower bound in [Goldreich et al., 2015] and [Cheng and Li, 2018]. We also generalize the main lower bound of [Goldreich et al., 2015] for extractors in AC⁰, showing that when k < n/poly log n, even strong dispersers do not exist in non-uniform AC⁰. For the NC¹ setting, we also give a logspace-uniform extractor construction with seed length O(log n/(ε)) and a small constant fraction entropy loss in the output. It works for every k ≥ O(log² n), ε ≥ 2^{-O(√k)}. Our main techniques include a new error reduction process and a new output stretch process, based on low-depth circuit implementations for mergers, condensers, and somewhere extractors.
Cite as

Kuan Cheng and Ruiyang Wu. Randomness Extractors in AC⁰ and NC¹: Optimal up to Constant Factors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 69:1-69:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cheng_et_al:LIPIcs.APPROX/RANDOM.2024.69,
  author =	{Cheng, Kuan and Wu, Ruiyang},
  title =	{{Randomness Extractors in AC⁰ and NC¹: Optimal up to Constant Factors}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{69:1--69:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.69},
  URN =		{urn:nbn:de:0030-drops-210623},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.69},
  annote =	{Keywords: randomness extractor, uniform AC⁰, error reduction, uniform NC¹, disperser}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.75
Derandomizing Multivariate Polynomial Factoring for Low Degree Factors

Authors: Pranjal Dutta, Amit Sinhababu, and Thomas Thierauf

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
Kaltofen [STOC 1986] gave a randomized algorithm to factor multivariate polynomials given by algebraic circuits. We derandomize the algorithm in some special cases. For an n-variate polynomial f of degree d from a class 𝒞 of algebraic circuits, we design a deterministic algorithm to find all its irreducible factors of degree ≤ δ, for constant δ. The running time of this algorithm stems from a deterministic PIT algorithm for class 𝒞 and a deterministic algorithm that tests divisibility of f by a polynomial of degree ≤ δ. By using the PIT algorithm for constant-depth circuits by Limaye, Srinivasan and Tavenas [FOCS 2021] and the divisibility results by Forbes [FOCS 2015], this generalizes and simplifies a recent result by Kumar, Ramanathan and Saptharishi [SODA 2024]. They designed a subexponential-time algorithm that, given a blackbox access to f computed by a constant-depth circuit, outputs its irreducible factors of degree ≤ δ. When the input f is sparse, the time complexity of our algorithm depends on a whitebox PIT algorithm for ∑_i m_i g_i^{d_i}, where m_i are monomials and deg(g_i) ≤ δ. All the previous algorithms required a blackbox PIT algorithm for the same class. Our second main result considers polynomials f, where each irreducible factor has degree at most δ. We show that all the irreducible factors with their multiplicities can be computed in polynomial time with blackbox access to f. Finally, we consider factorization of sparse polynomials. We show that in order to compute all the sparse irreducible factors efficiently, it suffices to derandomize irreducibility preserving bivariate projections for sparse polynomials.
Cite as

Pranjal Dutta, Amit Sinhababu, and Thomas Thierauf. Derandomizing Multivariate Polynomial Factoring for Low Degree Factors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 75:1-75:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dutta_et_al:LIPIcs.APPROX/RANDOM.2024.75,
  author =	{Dutta, Pranjal and Sinhababu, Amit and Thierauf, Thomas},
  title =	{{Derandomizing Multivariate Polynomial Factoring for Low Degree Factors}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{75:1--75:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.75},
  URN =		{urn:nbn:de:0030-drops-210687},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.75},
  annote =	{Keywords: algebraic complexity, factoring, low degree, weight isolation, divisibility}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.9
Quantum Algorithms for Hopcroft’s Problem

Authors: Vladimirs Andrejevs, Aleksandrs Belovs, and Jevgēnijs Vihrovs

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
In this work we study quantum algorithms for Hopcroft’s problem which is a fundamental problem in computational geometry. Given n points and n lines in the plane, the task is to determine whether there is a point-line incidence. The classical complexity of this problem is well-studied, with the best known algorithm running in O(n^{4/3}) time, with matching lower bounds in some restricted settings. Our results are two different quantum algorithms with time complexity Õ(n^{5/6}). The first algorithm is based on partition trees and the quantum backtracking algorithm. The second algorithm uses a quantum walk together with a history-independent dynamic data structure for storing line arrangement which supports efficient point location queries. In the setting where the number of points and lines differ, the quantum walk-based algorithm is asymptotically faster. The quantum speedups for the aforementioned data structures may be useful for other geometric problems.
Cite as

Vladimirs Andrejevs, Aleksandrs Belovs, and Jevgēnijs Vihrovs. Quantum Algorithms for Hopcroft’s Problem. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{andrejevs_et_al:LIPIcs.MFCS.2024.9,
  author =	{Andrejevs, Vladimirs and Belovs, Aleksandrs and Vihrovs, Jevg\={e}nijs},
  title =	{{Quantum Algorithms for Hopcroft’s Problem}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{9:1--9:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.9},
  URN =		{urn:nbn:de:0030-drops-205653},
  doi =		{10.4230/LIPIcs.MFCS.2024.9},
  annote =	{Keywords: Quantum algorithms, Quantum walks, Computational Geometry}
}
Document
DOI: 10.4230/LIPIcs.SAT.2024.23
eSLIM: Circuit Minimization with SAT Based Local Improvement

Authors: Franz-Xaver Reichl, Friedrich Slivovsky, and Stefan Szeider

Published in: LIPIcs, Volume 305, 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)

Abstract
eSLIM is a tool for circuit minimization that utilizes Exact Synthesis and the SAT-based local improvement method (SLIM) to locally improve circuits. eSLIM improves upon the earlier prototype CIOPS that uses Quantified Boolean Formulas (QBF) to succinctly encode resynthesis of multi-output subcircuits subject to don't cares. This paper describes two improvements. First, it presents a purely propositional encoding based on a Boolean relation characterizing the input-output behavior of the subcircuit under don't cares. This allows the use of a SAT solver for resynthesis, substantially reducing running times when applied to functions from the IWLS 2023 competition, where eSLIM placed second. Second, it proposes circuit partitioning techniques in which don't cares for a subcircuit are captured only with respect to an enclosing window, rather than the entire circuit. Circuit partitioning trades completeness for efficiency, and successfully enables the application of exact synthesis to some of the largest circuits in the EPFL suite, leading to improvements over the current best implementation for several instances.
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Franz-Xaver Reichl, Friedrich Slivovsky, and Stefan Szeider. eSLIM: Circuit Minimization with SAT Based Local Improvement. In 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 305, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{reichl_et_al:LIPIcs.SAT.2024.23,
  author =	{Reichl, Franz-Xaver and Slivovsky, Friedrich and Szeider, Stefan},
  title =	{{eSLIM: Circuit Minimization with SAT Based Local Improvement}},
  booktitle =	{27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)},
  pages =	{23:1--23:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-334-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{305},
  editor =	{Chakraborty, Supratik and Jiang, Jie-Hong Roland},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2024.23},
  URN =		{urn:nbn:de:0030-drops-205458},
  doi =		{10.4230/LIPIcs.SAT.2024.23},
  annote =	{Keywords: QBF, Exact Synthesis, Circuit Minimization, SLIM}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.5
Explicit Time and Space Efficient Encoders Exist Only with Random Access

Authors: Joshua Cook and Dana Moshkovitz

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
We give the first explicit constant rate, constant relative distance, linear codes with an encoder that runs in time n^{1 + o(1)} and space polylog(n) provided random access to the message. Prior to this work, the only such codes were non-explicit, for instance repeat accumulate codes [Divsalar et al., 1998] and the codes described in [Gál et al., 2013]. To construct our codes, we also give explicit, efficiently invertible, lossless condensers with constant entropy gap and polylogarithmic seed length. In contrast to encoders with random access to the message, we show that encoders with sequential access to the message can not run in almost linear time and polylogarithmic space. Our notion of sequential access is much stronger than streaming access.
Cite as

Joshua Cook and Dana Moshkovitz. Explicit Time and Space Efficient Encoders Exist Only with Random Access. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 5:1-5:54, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cook_et_al:LIPIcs.CCC.2024.5,
  author =	{Cook, Joshua and Moshkovitz, Dana},
  title =	{{Explicit Time and Space Efficient Encoders Exist Only with Random Access}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{5:1--5:54},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.5},
  URN =		{urn:nbn:de:0030-drops-204015},
  doi =		{10.4230/LIPIcs.CCC.2024.5},
  annote =	{Keywords: Time-Space Trade Offs, Error Correcting Codes, Encoders, Explicit Constructions, Streaming Lower Bounds, Sequential Access, Time-Space Lower Bounds, Lossless Condensers, Invertible Condensers, Condensers}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.8
Asymptotically-Good RLCCs with (log n)^(2+o(1)) Queries

Authors: Gil Cohen and Tal Yankovitz

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
Recently, Kumar and Mon reached a significant milestone by constructing asymptotically good relaxed locally correctable codes (RLCCs) with poly-logarithmic query complexity. Specifically, they constructed n-bit RLCCs with O(log^{69} n) queries. Their construction relies on a clever reduction to locally testable codes (LTCs), capitalizing on recent breakthrough works in LTCs. As for lower bounds, Gur and Lachish (SICOMP 2021) proved that any asymptotically-good RLCC must make Ω̃(√{log n}) queries. Hence emerges the intriguing question regarding the identity of the least value 1/2 ≤ e ≤ 69 for which asymptotically-good RLCCs with query complexity (log n)^{e+o(1)} exist. In this work, we make substantial progress in narrowing the gap by devising asymptotically-good RLCCs with a query complexity of (log n)^{2+o(1)}. The key insight driving our work lies in recognizing that the strong guarantee of local testability overshoots the requirements for the Kumar-Mon reduction. In particular, we prove that we can replace the LTCs by "vanilla" expander codes which indeed have the necessary property: local testability in the code’s vicinity.
Cite as

Gil Cohen and Tal Yankovitz. Asymptotically-Good RLCCs with (log n)^(2+o(1)) Queries. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cohen_et_al:LIPIcs.CCC.2024.8,
  author =	{Cohen, Gil and Yankovitz, Tal},
  title =	{{Asymptotically-Good RLCCs with (log n)^(2+o(1)) Queries}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{8:1--8:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.8},
  URN =		{urn:nbn:de:0030-drops-204045},
  doi =		{10.4230/LIPIcs.CCC.2024.8},
  annote =	{Keywords: Relaxed locally decodable codes, Relxaed locally correctable codes, RLCC, RLDC}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.17
Local Enumeration and Majority Lower Bounds

Authors: Mohit Gurumukhani, Ramamohan Paturi, Pavel Pudlák, Michael Saks, and Navid Talebanfard

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
Depth-3 circuit lower bounds and k-SAT algorithms are intimately related; the state-of-the-art Σ^k_3-circuit lower bound (Or-And-Or circuits with bottom fan-in at most k) and the k-SAT algorithm of Paturi, Pudlák, Saks, and Zane (J. ACM'05) are based on the same combinatorial theorem regarding k-CNFs. In this paper we define a problem which reveals new interactions between the two, and suggests a concrete approach to significantly stronger circuit lower bounds and improved k-SAT algorithms. For a natural number k and a parameter t, we consider the Enum(k, t) problem defined as follows: given an n-variable k-CNF and an initial assignment α, output all satisfying assignments at Hamming distance t(n) of α, assuming that there are no satisfying assignments of Hamming distance less than t(n) of α. We observe that an upper bound b(n, k, t) on the complexity of Enum(k, t) simultaneously implies depth-3 circuit lower bounds and k-SAT algorithms: - Depth-3 circuits: Any Σ^k_3 circuit computing the Majority function has size at least binom(n,n/2)/b(n, k, n/2). - k-SAT: There exists an algorithm solving k-SAT in time O(∑_{t=1}^{n/2}b(n, k, t)). A simple construction shows that b(n, k, n/2) ≥ 2^{(1 - O(log(k)/k))n}. Thus, matching upper bounds for b(n, k, n/2) would imply a Σ^k_3-circuit lower bound of 2^Ω(log(k)n/k) and a k-SAT upper bound of 2^{(1 - Ω(log(k)/k))n}. The former yields an unrestricted depth-3 lower bound of 2^ω(√n) solving a long standing open problem, and the latter breaks the Super Strong Exponential Time Hypothesis. In this paper, we propose a randomized algorithm for Enum(k, t) and introduce new ideas to analyze it. We demonstrate the power of our ideas by considering the first non-trivial instance of the problem, i.e., Enum(3, n/2). We show that the expected running time of our algorithm is 1.598ⁿ, substantially improving on the trivial bound of 3^{n/2} ≃ 1.732ⁿ. This already improves Σ^3_3 lower bounds for Majority function to 1.251ⁿ. The previous bound was 1.154ⁿ which follows from the work of Håstad, Jukna, and Pudlák (Comput. Complex.'95). By restricting ourselves to monotone CNFs, Enum(k, t) immediately becomes a hypergraph Turán problem. Therefore our techniques might be of independent interest in extremal combinatorics.
Cite as

Mohit Gurumukhani, Ramamohan Paturi, Pavel Pudlák, Michael Saks, and Navid Talebanfard. Local Enumeration and Majority Lower Bounds. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 17:1-17:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gurumukhani_et_al:LIPIcs.CCC.2024.17,
  author =	{Gurumukhani, Mohit and Paturi, Ramamohan and Pudl\'{a}k, Pavel and Saks, Michael and Talebanfard, Navid},
  title =	{{Local Enumeration and Majority Lower Bounds}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{17:1--17:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.17},
  URN =		{urn:nbn:de:0030-drops-204136},
  doi =		{10.4230/LIPIcs.CCC.2024.17},
  annote =	{Keywords: Depth 3 circuits, k-CNF satisfiability, Circuit lower bounds, Majority function}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.18
Pseudorandomness, Symmetry, Smoothing: I

Authors: Harm Derksen, Peter Ivanov, Chin Ho Lee, and Emanuele Viola

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
We prove several new results about bounded uniform and small-bias distributions. A main message is that, small-bias, even perturbed with noise, does not fool several classes of tests better than bounded uniformity. We prove this for threshold tests, small-space algorithms, and small-depth circuits. In particular, we obtain small-bias distributions that - achieve an optimal lower bound on their statistical distance to any bounded-uniform distribution. This closes a line of research initiated by Alon, Goldreich, and Mansour in 2003, and improves on a result by O'Donnell and Zhao. - have heavier tail mass than the uniform distribution. This answers a question posed by several researchers including Bun and Steinke. - rule out a popular paradigm for constructing pseudorandom generators, originating in a 1989 work by Ajtai and Wigderson. This again answers a question raised by several researchers. For branching programs, our result matches a bound by Forbes and Kelley. Our small-bias distributions above are symmetric. We show that the xor of any two symmetric small-bias distributions fools any bounded function. Hence our examples cannot be extended to the xor of two small-bias distributions, another popular paradigm whose power remains unknown. We also generalize and simplify the proof of a result of Bazzi.
Cite as

Harm Derksen, Peter Ivanov, Chin Ho Lee, and Emanuele Viola. Pseudorandomness, Symmetry, Smoothing: I. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 18:1-18:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{derksen_et_al:LIPIcs.CCC.2024.18,
  author =	{Derksen, Harm and Ivanov, Peter and Lee, Chin Ho and Viola, Emanuele},
  title =	{{Pseudorandomness, Symmetry, Smoothing: I}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{18:1--18:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.18},
  URN =		{urn:nbn:de:0030-drops-204144},
  doi =		{10.4230/LIPIcs.CCC.2024.18},
  annote =	{Keywords: pseudorandomness, k-wise uniform distributions, small-bias distributions, noise, symmetric tests, thresholds, Krawtchouk polynomials}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.20
Lower Bounds for Set-Multilinear Branching Programs

Authors: Prerona Chatterjee, Deepanshu Kush, Shubhangi Saraf, and Amir Shpilka

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
In this paper, we prove super-polynomial lower bounds for the model of sum of ordered set-multilinear algebraic branching programs, each with a possibly different ordering (∑smABP). Specifically, we give an explicit nd-variate polynomial of degree d such that any ∑smABP computing it must have size n^ω(1) for d as low as ω(log n). Notably, this constitutes the first such lower bound in the low degree regime. Moreover, for d = poly(n), we demonstrate an exponential lower bound. This result generalizes the seminal work of Nisan (STOC, 1991), which proved an exponential lower bound for a single ordered set-multilinear ABP. The significance of our lower bounds is underscored by the recent work of Bhargav, Dwivedi, and Saxena (TAMC, 2024), which showed that super-polynomial lower bounds against a sum of ordered set-multilinear branching programs - for a polynomial of sufficiently low degree - would imply super-polynomial lower bounds against general ABPs, thereby resolving Valiant’s longstanding conjecture that the permanent polynomial can not be computed efficiently by ABPs. More precisely, their work shows that if one could obtain such lower bounds when the degree is bounded by O(log n/ log log n), then it would imply super-polynomial lower bounds against general ABPs. Our results strengthen the works of Arvind & Raja (Chic. J. Theor. Comput. Sci., 2016) and Bhargav, Dwivedi & Saxena (TAMC, 2024), as well as the works of Ramya & Rao (Theor. Comput. Sci., 2020) and Ghoshal & Rao (International Computer Science Symposium in Russia, 2021), each of which established lower bounds for related or restricted versions of this model. They also strongly answer a question from the former two, which asked to prove super-polynomial lower bounds for general ∑smABP.
Cite as

Prerona Chatterjee, Deepanshu Kush, Shubhangi Saraf, and Amir Shpilka. Lower Bounds for Set-Multilinear Branching Programs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2024.20,
  author =	{Chatterjee, Prerona and Kush, Deepanshu and Saraf, Shubhangi and Shpilka, Amir},
  title =	{{Lower Bounds for Set-Multilinear Branching Programs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{20:1--20:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.20},
  URN =		{urn:nbn:de:0030-drops-204167},
  doi =		{10.4230/LIPIcs.CCC.2024.20},
  annote =	{Keywords: Lower Bounds, Algebraic Branching Programs, Set-multilinear polynomials}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.31
Low-Depth Algebraic Circuit Lower Bounds over Any Field

Authors: Michael A. Forbes

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
The recent breakthrough of Limaye, Srinivasan and Tavenas [Limaye et al., 2022] (LST) gave the first super-polynomial lower bounds against low-depth algebraic circuits, for any field of zero (or sufficiently large) characteristic. It was an open question to extend this result to small-characteristic ([Limaye et al., 2022; Govindasamy et al., 2022; Fournier et al., 2023]), which in particular is relevant for an approach to prove superpolynomial AC⁰[p]-Frege lower bounds ([Govindasamy et al., 2022]). In this work, we prove super-polynomial algebraic circuit lower bounds against low-depth algebraic circuits over any field, with the same parameters as LST (or even matching the improved parameters of Bhargav, Dutta, and Saxena [Bhargav et al., 2022]). We give two proofs. The first is logical, showing that even though the proof of LST naively fails in small characteristic, the proof is sufficiently algebraic that generic transfer results imply the result over characteristic zero implies the result over all fields. Motivated by this indirect proof, we then proceed to give a second constructive proof, replacing the field-dependent set-multilinearization result of LST with a set-multilinearization that works over any field, by using the Binet-Minc identity [Minc, 1979].
Cite as

Michael A. Forbes. Low-Depth Algebraic Circuit Lower Bounds over Any Field. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{forbes:LIPIcs.CCC.2024.31,
  author =	{Forbes, Michael A.},
  title =	{{Low-Depth Algebraic Circuit Lower Bounds over Any Field}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{31:1--31:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.31},
  URN =		{urn:nbn:de:0030-drops-204271},
  doi =		{10.4230/LIPIcs.CCC.2024.31},
  annote =	{Keywords: algebraic circuits, lower bounds, low-depth circuits, positive characteristic}
}
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